Entailment of Type Equalities

Our aim is to derive the entailment judgement

g1, .., gn |- w1, .., wm

i.e., whether we can derive the wanted equalities w1 to wm from the given equalities g1, .., gn under a given set of toplevel equality schemas (i.e., equalities involving universally quantified variables). We permit unification variables in the wanted equalities, and a derivation may include the instantiation of these variables; i.e., may compute a unifier. However, that unifer must be most general.

The derivation algorithm is complicated by the pragmatic requirement that, even if there is no derivation for the judgement, we would like to compute a unifier. This unifier should be as specific as possible under some not yet specified (strongest) weakening of the judgement so that it is derivable. (Whatever that means...)

The following is based on ideas for the new, post-ICFP'08 solving algorithm described in CVS papers/type-synonym/new-single.tex. A revised version of new-single.tex that integrates the core ideas from this wiki page is in papers/type-synonym/normalised_equations_algorithm.tex. Most of the code is in the module TcTyFuns.

Terminology

Wanted equality

An equality constraint that we need to derive during type checking. Failure to derive it leads to rejection of the checked program.

Local equality, given equality

An equality constraint that -in a certain scope- may be used to derive wanted equalities.

Flexible type variable, unification variable, HM variable

Type variables that may be globally instantiated by unification. We use Greek letters alpha, beta,... as names for these variables.

Rigid type variable, skolem type variable

Type variable that cannot be globally instantiated, but it may be locally refined by a local equality constraint. We use Roman letters a, b,... as names for these variables.

In positions where we can have both flexible and rigid variables, we use x, y, z.

Overall algorithm

The overall structure is as in new-single.tex, namely

normalise all constraints (both locals and wanteds),

solve the wanteds, and

finalise.

However, the three phases differ in important ways. In particular, normalisation includes decompositions & the occurs check, and we don't instantiate any flexible type variables before we finalise (i.e., solving is purely local).

Normal equalities

Central to the algorithm are normal equalities, which can be regarded as a set of rewrite rules. Normal equalities are carefully oriented and contain synonym families only as the head symbols of left-hand sides. They assume one of the following two major forms:

Family equality:co :: F t1..tn ~ t or

Variable equality:co :: x ~ t, where we again distinguish two forms:

Variable-term equality:co :: x ~ t, where t is not a variable, or

Variable-variable equality:co :: x ~ y, where x > y.

where

the types t, t1, ..., tn may not contain any occurrences of synonym families,

the left-hand side of an equality may not occur in the right-hand side, and

the relation x > y is an arbitrary, but total order on type variables.

The second bullet of the where clause is trivially true for equalities of Form (1); it also implies that the left- and right-hand sides are different.

Observations

Normal equalities are similar to equalities meeting the Orientation Invariant and Flattening Invariant of new-single, but they are not the same.

Normal equalities are never self-recursive. They can be mutually recursive. A mutually recursive group will exclusively contain variable equalities.

Coercions

Coercions co are either wanteds (represented by a flexible type variable) or givens aka locals (represented by a type term of kind CO). In GHC, they are represented by TcRnTypes.EqInstCo, which is defined as

type EqInstCo = Either
TcTyVar -- case for wanteds (variable to be filled with a witness)
Coercion -- case for locals

Whenever an equality of Form (2) or (3) would be recursive, the program can be rejected on the basis of a failed occurs check. (Immediate rejection is always justified, as right-hand sides do not contain synonym familles; hence, any recursive occurrences of a left-hand side imply that the equality is unsatisfiable.)

For all local equalities, we RECORD substitutions for fresh flexibles introduced by flatten by already updating the meta type variable. However, the update will only have an effect after the Insts have been zonked; i.e., after finalisation.

We drop all loopy equalities (see Section 8.2 of our ICFP'08 paper). If we drop a wanted, it is a type error; if we drop a local, we emit a warning (as we only sacrifice completeness).

We do the essentially same for class constraints; i.e., we use flatten to extract all family applications as equality constraints. However, instead of adjusting the coercion at the flattened constraint, we generate a dictionary binding for the new dictionary (using a cast expression).

Propagation (aka Solving)

Propagation is based on four rules that transform family and variable equalities. The first rule transforms a family equality using a toplevel equality schema. The other three use one equality to transform another. In the presentation, the transforming equality is always first and the transformed is second, separate by an ampersand (&).

Below the rules are two scheme for applying the rules. The first one naively iterates until the system doesn't change anymore. The second scheme optimises the iteration somewhat.

NB: Afterwards, we need to normalise co2'. Then, the SubstVarVar or SubstVarFam rules may apply to the results of normalisation. Moreover, co1 may have a SubstFam or a SubstVarFam match with rules other than the results of normalising co2'.

NB: Afterwards, we need to normalise co2'. Then, the SubstVarVar or SubstVarFam rules may apply to the results of normalisation, but not with co1, as s and t cannot contain x -- cf. the definition of normal equalities. However, co1 may have another SubstVarVar or SubstVarFam match with rules other than the results of normalising co2'.

where x occurs in F s1..sn, and co1 may be a wanted only if co2 is a wanted, too.

NB: No normalisation required. Afterwards, SubstVarVar or SubstVarFam may apply to co1 and all rules, except SubstVarVar, may apply to co2'. However, SubstVarFam cannot again apply to co1 and co2', as t cannot contain x -- cf. the definition of normal equalities.

NB: In the three substitution rules, it is never necessary to try using co1 with any of the equalities derived from co2'. Hence, after having considered one equality as co1 with every other equality, we can immediately put co1 into the list of residual equalities.

Rule application: specification

Propagation proceeds by applying any of the four rules until the system does not change anymore. After application of a rule, the equalities that were modified need to be normalised again:

Propagate = fix (Top | SubstFam | SubstVarVar | SubstVarFam)

Rule application: algorithm

The four propagation rules are implemented by the following four functions:

For applySubstFam, applySubstVarVar, and applySubstVarFam, if the rule is not applicable to the arguments in the given order, but it is applicable in the opposite order, return the second argument as if it had been modified. (As a result, it will get onto the todo list, and the equalities eventually be used in the opposite order.)

The following function gets a list with all local and wanted equalities. It returns a list of residual equalities that permit no further rule application.

Observations

Only SubstVarFam when replacing a variable in a family equality can lead to recursion with (Top).

A significant difference to new-single is that solving is a purely local operation. We never instantiate any flexible variables.

We cannot discard any variable equalities after substitution since we do not substitute into right-hand sides anymore. Moreover, in the concrete implementation, variables may also be re-introduced due to simplification of type classes (which currently doesn't happen in the same iteration).

Finalisation

The finalisation step instantiates as many flexible type variables as possible, but it takes care not to instantiate variables occurring in the global environment with types containing synonym family applications. This is important to obtain principle types (c.f., Andrew Kennedy's thesis). We perform finalisation in two phases:

Substitution:

Pass A: For any variable equality of the form co :: x ~ t (both local and wanted), we apply the substitution [t/x] to the right-hand side of all equalities. We also perform the same substitution on class constraints.

Pass B: Unless we are in inference mode, for any wanted family equality of the form co :: F t1..tn ~ alpha, we apply the substitution [F t1..tn/alpha] to both sides of all family equalities. We need to substitute all flexibles that arose as skolems during flattening of wanteds before we substitute any other flexibles.

Instantiation: For any variable equality of the form co :: alpha ~ t or co :: a ~ alpha, where co is wanted, we instantiate alpha with t or a, respectively, and set co := id. Moreover, we have to do the same for equalities of the form co :: F t1..tn ~ alpha unless we are in inference mode and alpha appears in the environment or any other wanteds. (We never instantiate any flexibles introduced by flattening locals.)

Important points are the following:

The substitution step can lead to recursive equalities; i.e., we need to apply an occurs check after each substitution.

We need to substitute all flexibles that arose as skolems during flattening of wanteds before we substitute any other flexibles. Consider F delta ~ alpha, F alpha ~ delta, where alpha is a skolem and delta a free flexible. We need to produce F (F delta) ~ delta (and not F (F alpha) ~ alpha). Otherwise, we may wrongly claim to having performed an improvement, which can lead to non-termination of the combined class-family solver.

We need to substitute family equalities into both sides of family equalities; consider, F t1..tn ~ alpha, G s1..sm ~ alpha.

Note that it is an important property of propagation that we only need to substitute into right-hand sides during finalisation. After finalisation and zonking all flattening of locals is undone (c.f., note below the flattening code above).

TODO If we use flexible variables for the flattening of the wanteds, too, the equality corresponding to x ~ a above will be oriented the other way around. That can be a problem because of the asymmetry of the SubstVar and SubstFun rules (i.e., wanted equalities are not substituted into locals).

Termination

SkolemOccurs

The Note [skolemOccurs loop] in the old code explains that equalities of the form x ~ t (where x is a flexible type variable) may not be used as rewrite rules, but only be solved by applying Rule Unify. As Unify carefully avoids cycles, this prevents the use of equalities introduced by the Rule SkolemOccurs as rewrite rules. For this to work, SkolemOccurs also had to apply to equalities of the form a ~ t[[a]]. This was a somewhat intricate set up that we seek to simplify here. Whether equalities of the form x ~ t are used as rewrite rules or solved by Unify doesn't matter anymore. Instead, we disallow recursive equalities after normalisation completely (both locals and wanteds). This is possible as right-hand sides are free of synonym families.

To look at this in more detail, let's consider the following notorious example:

E_t: forall x. F [x] ~ [F x]
[F v] ~ v ||- [F v] ~ v

New-single: The following derivation shows how the algorithm in new-single fails to terminate for this example.

New-single using flexible tyvars to flatten locals, but w/o Rule (Local) for flexible type variables: With (SkolemOccurs) it is crucial to avoid using Rule (Local) with flexible type variables. We can achieve a similar effect with new-single if we (a) use flexible type variables to flatten local equalities and (b) at the same time do not use Rule (Local) for variable equalities with flexible type variables. NB: Point (b) was necessary for the ICFP'08 algorithm, too.

A serious disadvantage of this approach is that we do want to use Rule (Local) with flexible type variables as soon as we have rank-n signatures. In fact, the lack of doing so is responsible for a few failing tests in the testsuite in the GHC implementation of (SkolemOccurs).

De-prioritise Rule (Local): Instead of outright forbidding the use of Rule (Local) with flexible type variables, we can simply require that Local is only used if no other rule is applicable. (That has the same effect on satisfiable queries, and in particular, the present example.)